From: Brett Paatsch (bpaatsch@bigpond.net.au)
Date: Wed Sep 10 2003 - 21:44:42 MDT
Eliezer S. Yudkowsky writes:
> Suppose that a bowl has 5 red chips and 3 white chips.
> We sample chips from the bowl using the following
> procedure: On each round we draw a random chip,
> replace it, and then add another chip of the same color
> to the bowl. For example, if on the first round we
> happen to draw a red chip, there would then be 6 red
> chips and 3 white chips to draw from on the second
> round.
>
> Given that a white chip was drawn on the fourth round,
> what is the probability that a white chip was drawn on the
> second round?
>
> (This problem is extra bonus evil because it's so easy if
> you know the rules.)
Well here's one would-be Bayesian who is ready to get the
"evil trick" explained.
I *loved* probability at secondary school as it seemed so
eminently useful and Bayes creates the same impression but
I find that I am frustrated by it in the same way.
Whilst I had a good intuitive handle on probability I had a
lot of trouble translating English language logic into the
mathematical formalisms that facilitate easier handling.
It was like putting the terms into mathematically formal
language was a language translation that I couldn't easily
do.
Intuitively I can see that at round two there are nine chips
total and either 3 or 4 of those 9 are white. Because of the
rules that mean an early white increases the number of whites
latter I can see that at the point when their are 11 chips in
round 4, the fact of a white in round 2 would mean more
of the 11 were likely to be white AND vice versa. I.e.. Given
a white IS chosen from the 11 the chips in round 4, this
probability tells us something about the likelihood that a white
was drawn in round 2 (it is higher than an even chance) but
that's where the wheels come off for me. I can't yet put these
facts into formal mathematical language (at least I think that's
my problem). I have the same problem with the formal rules
in Godel, Esher and Bach. Taking English language logic to
mathematical formalism and getting the benefits of being able
to manipulate the symbols is where I struggle.
I did take a look at your excruciatingly gentle introduction
to Bayes, (I liked it. It helped, and is worth a closer look)
but for now I' m ready to see the evil trick when you are
ready to reveal it ;-)
Regards,
Brett
(A keen but sorry maths student :-( )
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